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Encyclopaedia of DesignTheory: Latin squares

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Miscellanea about Latin squares

Classification

Brendan McKay has lists of representatives of the isomorphism types of Latin squares of order up to 8 under two natural notion of isomorphism: isotopy classes, where we are allowed to permute the rows, columns and symbols; and main classes, where we are also allowed to permute among themselves the three factors "rows", "columns" and "symbols". He also gives representatives of isotopy and main classes for the squares of order 9 with non-trivial automorphism groups (with the corresponding notions of automorphism).

The numbers of classes are given in the following table.

Order2345 678
Isotopy classes1122 225641676267
Main classes1122 12147283657

Random Latin squares

In the paper
M. T. Jacobson and P. Matthews, Generating uniformly distributed random Latin squares, J. Combinatorial Design 4 (1996), 405-437
there is a Markov chain method for generating a random Latin square of given order. The limiting distribution of the Markov chain makes all Latin squares equally likely. However, little is known about the rate of convergence.

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Peter J. Cameron
2 August 2002